Discrete eigenvalue optimization from entropic smoothing and first-order methods

What the math gives to ML

The paper formulates combinatorial selection through the smallest eigenvalue of an affine positive-semidefinite matrix, with M(x)=C+sum_i x_i m_i m_i^T and binary decisions x_i. This objective rewards selections whose vectors span all directions well, rather than merely selecting high-norm or individually strong candidates. A direct neural-network transfer is spectral-diversity pruning or expert/channel selection: construct candidate activation signatures m_i, relax binary selection to a capped simplex, optimize a differentiable entropic soft minimum eigenvalue, then round to the desired budget. The resulting module should retain complementary channels or experts and can improve accuracy at a fixed parameter, FLOP, or routing budget.